Vibration Analysis of Rectangular and Circular Plates with Fixed Edges: Analytical and Computational Validation

This study presents a comparative investigation of the vibration characteristics of rectangular and circular plates with fixed edges using analytical, numerical, and computational approaches. Analytical models based on classical plate theory were employed to calculate natural frequencies and mode shapes, while finite element analysis (FEA) was performed in a CAE tool to provide high-fidelity simulation results. A detailed mesh convergence study confirmed numerical stability, with frequency variations below 1% between successive refinements. Analytical predictions showed excellent agreement with simulation results for lower modes, with errors as low as 0.25% for the rectangular plate and 2.65% for the circular plate. However, higher modes exhibited significant deviations, with errors reaching up to 29.01% for rectangular and 181.52% for circular geometries, highlighting the limitations of closed-form solutions in capturing complex vibrational behavior. Python-based computational tools were developed to automate frequency calculations and visualize mode shapes, bridging theoretical formulations with practical applications. The integration of analytical, numerical, and computational methods provides a robust framework for understanding plate vibrations, demonstrating that analytical solutions remain effective for fundamental modes, while FEM ensures accuracy for higher-order behavior.